Revised Kempe-class conjecture for almost bipartite graphs

Let $G$ be a graph formed from a bipartite graph $B$ by adding $\ell$ edges, so that $G$ is a $(k-1)$-colorable $B+E_\ell$ graph. For a graph $G$ and positive integer $k$, let $\operatorname{Kc}(G,k)$ denote the number of equivalence classes of $k$-colorings, where two colorings are equivalent if one can be transformed into the other by a sequence of Kempe swaps. Let $\mathbf{1}_{k-1}$ denote the indicator of the relevant condition on $k-1$ as used in the source.

Revised Kempe-class conjecture. If $G$ is a $(k-1)$-colorable $B+E_\ell$ graph with $k\geq 4$ and

$$\ell<\frac{k^2+8k-45+\mathbf{1}_{k-1}}{4},$$

then

$$\operatorname{Kc}(G,k)=1.$$

This is proposed as a revised version after the original conjecture was disproved in part of its range. The supplied text does not state whether the revised conjecture has been proved or refuted.